Some
Notes on Avogadro's Number, 6.022 x 10^{23}
T.A. Furtsch, Tennessee Technological University, Cookeville
Chemists use Avogadro's number every
day. It is a very valuable number for a chemist to know how
to use, and use properly. Where did Avogadro's number come
from? Did Avogadro himself do all the
calculations? Was it just arbitrarily made
up? How can it be measured? Some possible
answers follow.
Amadeo Avogadro
(17761856) was the author of Avogadro's Hypothesis in 1811,
which, together with GayLussac's Law of Combining Volumes,
was used by Stanislao Cannizzaro to elegantly
remove all doubt about the establishment of the atomic weight scale at
the Karlsruhe Conference of 1860.

The name
"Avogadro's Number" is just an honorary name
attached to the calculated value of the number of atoms, molecules,
etc. in a gram mole of any chemical substance. On the NIST
web site,
(National Institute of Standards and Technology, formerly the National
Bureau of Standards), it is now refered to as the Avogadro
Constant. Of course if we used some other mass unit for the
mole
such as "pound mole", the "number" would be different than 6.022 x 10^{23}.
The first
person to have calculated the number of molecules in any mass of
substance was Josef Loschmidt, (18211895), an
Austrian high school teacher, who in 1865, using the new Kinetic
Molecular Theory (KMT) calculated the number of molecules in one cubic
centimeter of gaseous substance under ordinary conditions of
temperature and pressure, to be somewhere around 2.6 x 10^{19}
molecules. This is usually known
as "Loschmidt's
Constant." (This value, n_{o}, is now
listed at the NIST
web site as 2.686 7774 x 10^{25} m^{3})

When was the
first time the term "Avogadro Number" was used? The
designation seems to originate in a 1909 paper entitled "Brownian
Movement and Molecular Reality." by Jean
Baptiste Jean Perrin
(b. Lille,
France, 30.9.1870d. New York, 17.4.1942.) This paper was
translated into English from the French in Annals De Chimie
et de Physique by Fredric Soddy and is available.
Perrin, was the 1926 Nobel Laureate in Physics
for his work on the discontinuous structure of matter, and especially
for his discovery of sedimentation equilibrium. Perrin should
be very well known to anyone who does calculations in molecular
dynamics. Most of these methods were developed by
Perrin. In his paper Perrin says "The invariable number N is a
universal constant, which may be appropriately designated "Avogadro's
Constant."
In the presentation
of his Nobel prize in 1926 it was said of the work of Perrin:
It may perhaps be said that
in the work which we have just summarized Perrin has offered indirect
evidence for the existence of molecules. Here, follows a direct
evidence. Microscopic particles in a liquid are never at rest. They are
in perpetual movement, even under conditions of perfect external
equilibrium, constant temperature, etc. The only irrefutable
explanation for this phenomenon ascribes the movements of the particles
to shocks produced on them by the molecules of the liquid themselves. A
mathematical theory of this phenomenon has been given by Einstein.
The first experimental proof of this theory was given by a German
physicist, Seddig. After him, the problem was taken up by two
scientists simultaneously. One of them was Perrin; the other Svedberg.
I have to speak of Perrin only. His measurements on the Brownian
movement showed that Einstein's theory was in perfect agreement with
reality. Through these measurements a new determination of Avogadro's
number was obtained.

The molecular impacts produce
not only a forward movement of the particles distributed in a liquid,
but also a rotational movement. The theory of this rotation was
developed by Einstein. Measurements in relation herewith were carried
out by Perrin. In these measurements he has found another method for
determining Avogadro's number. What then is the result of these
researches ? How many molecules are there in two grams of hydrogen? The
three methods have given the following answers to this question: 68.2 x
10^{22}; 68.8 x 10^{22}; 65 x 10^{22}.
The work of Einstein and
Perrin gave some of the first concrete evidence for the existence of
molecules, entities many still did not recognize even into the early
1900's. And Avogadro's Number has a value that must
be measured experimentally. Subsequent to
the work of Loschmidt and Perrin many scientists carried out many
experiments using a variety of techniques to arrive at the most
accurate value for this the number of molecules in one mole of
substance. And by 1933 there was still no universal agreement
as to what the number should be called. In a paper entitled "Loschmidt's
Number", published in 1933 (Science
Progress, v. 27,
1933, pp. 634649), S. E. Virgo, a physicist at The
University, Sheffield, England says:
This number is frequently
referred to as "Avogadro's Number," the term "Loschmidt's Number" being
then reserved for the number of molecules in a cubic centimetre of a
gas under standard conditions. Unfortunately, these designations are
often interchanged. Avogadro's important hypothesis on the identity of
the numbers of molecules in equal volumes of different gases at the
same pressure and temperature was formulated in 1811, and is
appropriately associated with his name; but Avogadro made no
quantitative estimate of either of the abovementioned constants. The
first actual estimate of the number of molecules in one cubic
centimetre of a gas under standard conditions was made in 1865 by
Loschmidt, and from this the number of molecules (atoms) in a gram
molecule (atom) was later evaluated. From the quantitative viewpoint
it thus seems preferable to speak of "Loschmidt's number per
grammolecule (atom)," and of "Loschmidt's number per cubic
centimetre," as is almost invariably done in the German scientific
literature. This terminology avoids ambiguity, and has been adopted
here.
So, even by 1933, there was
no clear agreement as to what the number should be
called. Virgo goes on to say that by that year more
than eighty separate determinations had been made to discover the true
value of the number "as it is a basic atomic constant its most probable
value is of great importance in atomic physics." The best
modern values for what we now call "Avogadro's Number" are the result
of the xray diffraction measurement of lattice distances in metals and
salts. The earliest attempts at using this method are reviewed in
Virgo's paper. Calculations reflecting these
methods are often found in modern general chemistry text books. For
example, from xray
data the one can determine that titanium (Ti) has a bodycentered cubic
unit cell (i.e.there are two Ti atoms per unit cell) and an edge length
of 330.6 pm. One can also find that the density of Ti metal
is 4.401 g/cm^{3}. The number
of atoms of Ti in a mole of Ti (47.88 g), Avogadro's Number, can be
calculated as follows: (General Chemistry, Whitten,
Davis and Peck, Saunders College Publishing, 6ed, 2000, p. 523.)
Today's best experimental
value of 6.022 141 79 x 10^{23}
mol^{1} atoms per mol (obtained from NIST
web site) is the best average for measurements using the best methods
available. The experiments are often very difficult to carry
out. That the number today has 9 significant figures is a testament to
the quality of modern experimental methods.
Some Links related to this
essay:
Avogadro's
1811 Essay in which he hypothesizes that equal volumes of
gases contain equal numbers of molecules.(from Carmen
Giunta's classical chemistry page)
"Loschmidt's
Number", Science Progress, v. 27, 1933, pp. 634649.
Avogadro's Hypothesis
A Biographical interview with Amadeo
Avogadro